Bernoulli and the St. Petersburg Paradox

Magister Ludi


The expected utility hypothesis stems from Daniel Bernoulli's (1738) solution to the famous St. Petersburg Paradox posed in 1713 by his cousin Nicholas Bernoulli (it is common to note that Gabriel Cramer, another Swiss mathematician, also provided effectively the same solution ten years before Bernoulli). The Paradox challenges the old idea that people value random ventures according to its expected return. The Paradox posed the following situation: a fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2n ducats. How much should one pay to play this game? The paradox, of course, is that the expected return is infinite, namely:

E(w) = ・/font> i=1 (1/2n)キ2n = (1/2)キ2 + (1/4)22 + (1/8)23 + .... = 1 + 1 + 1 + ..... =

Yet while the expected payoff is infinite, one would not suppose, at least intuitively, that real-world people would be willing to pay an infinite amount of money to play this!

Daniel Bernoulli's solution involved two ideas that have since revolutionized economics: firstly, that people's utility from wealth, u(w), is not linearly related to wealth (w) but rather increases at a decreasing rate - the famous idea of diminishing marginal utility, u (Y) > 0 and u (Y) < 0; (ii) that a person's valuation of a risky venture is not the expected return of that venture, but rather the expected utility from that venture. In the St. Petersburg case, the value of the game to an agent (assuming initial wealth is zero) is:

E(u) = ・/font> i=1 (1/2n)キu(2n) = (1/2)キu(2) + (1/4)キu(22) + (1/8)キu(23) + .... <

which Bernoulli conjectured is finite because of the principle of diminishing marginal utility. (Bernoulli originally used a logarithmic function of the type u(x) = a log x). Consequently, people would only be willing to pay a finite amount of money to play this, even though its expected return is infinite. In general, by Bernoulli's logic, the valuation of any risky venture takes the expected utility form:

E(u | p, X) = ・/font> x X p(x)u(x)

where X is the set of possible outcomes, p(x) is the probability of a particular outcome x X and u: X R is a utility function over outcomes.

[Note: as Karl Menger (1934) later pointed out, placing an ironical twist on all this, Bernoulli's hypothesis of diminishing marginal utility is actually not enough to solve all St. Petersburg-type Paradoxes. To see this, note that we can always find a sequence of payoffs x1, x2, x3, .., which yield infinite expected value, and then propose, say, that u(xn) = 2n so that expected utility is also infinite. Thus, Menger proposed that utility must also be bounded above for paradoxes of this type to be resolved.]

Channelled by Gossen (1854), Bernoulli's idea of diminishing marginal utility of wealth became a centerpiece in the Marginalist Revolution of 1871-4 in the work of Jevons (1871), Menger (1871) and Walras (1874). However, Bernoulli's expected utility hypothesis has a thornier history. With only a handful of exceptions (e.g. Marshall, 1890: pp.111-2, 693-4; Edgeworth, 1911), it was never really picked up until John von Neumann and Oskar Morgenstern's (1944) Theory of Games and Economic Behavior, which we turn to next.

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